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  <h3><a href="../contents.html">Table of Contents</a></h3>
  <ul>
<li><a class="reference internal" href="#">Discrete Fourier Transform (<code class="xref py py-mod docutils literal notranslate"><span class="pre">numpy.fft</span></code>)</a><ul>
<li><a class="reference internal" href="#standard-ffts">Standard FFTs</a></li>
<li><a class="reference internal" href="#real-ffts">Real FFTs</a></li>
<li><a class="reference internal" href="#hermitian-ffts">Hermitian FFTs</a></li>
<li><a class="reference internal" href="#helper-routines">Helper routines</a></li>
<li><a class="reference internal" href="#background-information">Background information</a></li>
<li><a class="reference internal" href="#implementation-details">Implementation details</a></li>
<li><a class="reference internal" href="#type-promotion">Type Promotion</a></li>
<li><a class="reference internal" href="#normalization">Normalization</a></li>
<li><a class="reference internal" href="#real-and-hermitian-transforms">Real and Hermitian transforms</a></li>
<li><a class="reference internal" href="#higher-dimensions">Higher dimensions</a></li>
<li><a class="reference internal" href="#references">References</a></li>
<li><a class="reference internal" href="#examples">Examples</a></li>
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  <span class="target" id="module-numpy.fft"><span id="routines-fft"></span></span><div class="section" id="discrete-fourier-transform-numpy-fft">
<h1>Discrete Fourier Transform (<a class="reference internal" href="#module-numpy.fft" title="numpy.fft"><code class="xref py py-mod docutils literal notranslate"><span class="pre">numpy.fft</span></code></a>)<a class="headerlink" href="#discrete-fourier-transform-numpy-fft" title="Permalink to this headline">¶</a></h1>
<div class="section" id="standard-ffts">
<h2>Standard FFTs<a class="headerlink" href="#standard-ffts" title="Permalink to this headline">¶</a></h2>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fft.html#numpy.fft.fft" title="numpy.fft.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fft</span></code></a>(a[, n, axis, norm])</p></td>
<td><p>Compute the one-dimensional discrete Fourier Transform.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifft.html#numpy.fft.ifft" title="numpy.fft.ifft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifft</span></code></a>(a[, n, axis, norm])</p></td>
<td><p>Compute the one-dimensional inverse discrete Fourier Transform.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fft2.html#numpy.fft.fft2" title="numpy.fft.fft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fft2</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the 2-dimensional discrete Fourier Transform</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifft2.html#numpy.fft.ifft2" title="numpy.fft.ifft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifft2</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the 2-dimensional inverse discrete Fourier Transform.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fftn.html#numpy.fft.fftn" title="numpy.fft.fftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftn</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the N-dimensional discrete Fourier Transform.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifftn.html#numpy.fft.ifftn" title="numpy.fft.ifftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifftn</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the N-dimensional inverse discrete Fourier Transform.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="real-ffts">
<h2>Real FFTs<a class="headerlink" href="#real-ffts" title="Permalink to this headline">¶</a></h2>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a>(a[, n, axis, norm])</p></td>
<td><p>Compute the one-dimensional discrete Fourier Transform for real input.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.irfft.html#numpy.fft.irfft" title="numpy.fft.irfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft</span></code></a>(a[, n, axis, norm])</p></td>
<td><p>Compute the inverse of the n-point DFT for real input.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.rfft2.html#numpy.fft.rfft2" title="numpy.fft.rfft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft2</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the 2-dimensional FFT of a real array.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.irfft2.html#numpy.fft.irfft2" title="numpy.fft.irfft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft2</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the 2-dimensional inverse FFT of a real array.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.rfftn.html#numpy.fft.rfftn" title="numpy.fft.rfftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfftn</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the N-dimensional discrete Fourier Transform for real input.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.irfftn.html#numpy.fft.irfftn" title="numpy.fft.irfftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfftn</span></code></a>(a[, s, axes, norm])</p></td>
<td><p>Compute the inverse of the N-dimensional FFT of real input.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="hermitian-ffts">
<h2>Hermitian FFTs<a class="headerlink" href="#hermitian-ffts" title="Permalink to this headline">¶</a></h2>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.hfft.html#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a>(a[, n, axis, norm])</p></td>
<td><p>Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ihfft.html#numpy.fft.ihfft" title="numpy.fft.ihfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ihfft</span></code></a>(a[, n, axis, norm])</p></td>
<td><p>Compute the inverse FFT of a signal that has Hermitian symmetry.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="helper-routines">
<h2>Helper routines<a class="headerlink" href="#helper-routines" title="Permalink to this headline">¶</a></h2>
<table class="longtable docutils align-default">
<colgroup>
<col style="width: 10%" />
<col style="width: 90%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fftfreq.html#numpy.fft.fftfreq" title="numpy.fft.fftfreq"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftfreq</span></code></a>(n[, d])</p></td>
<td><p>Return the Discrete Fourier Transform sample frequencies.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.rfftfreq.html#numpy.fft.rfftfreq" title="numpy.fft.rfftfreq"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfftfreq</span></code></a>(n[, d])</p></td>
<td><p>Return the Discrete Fourier Transform sample frequencies (for usage with rfft, irfft).</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/numpy.fft.fftshift.html#numpy.fft.fftshift" title="numpy.fft.fftshift"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftshift</span></code></a>(x[, axes])</p></td>
<td><p>Shift the zero-frequency component to the center of the spectrum.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/numpy.fft.ifftshift.html#numpy.fft.ifftshift" title="numpy.fft.ifftshift"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifftshift</span></code></a>(x[, axes])</p></td>
<td><p>The inverse of <a class="reference internal" href="generated/numpy.fft.fftshift.html#numpy.fft.fftshift" title="numpy.fft.fftshift"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fftshift</span></code></a>.</p></td>
</tr>
</tbody>
</table>
</div>
<div class="section" id="background-information">
<h2>Background information<a class="headerlink" href="#background-information" title="Permalink to this headline">¶</a></h2>
<p>Fourier analysis is fundamentally a method for expressing a function as a
sum of periodic components, and for recovering the function from those
components.  When both the function and its Fourier transform are
replaced with discretized counterparts, it is called the discrete Fourier
transform (DFT).  The DFT has become a mainstay of numerical computing in
part because of a very fast algorithm for computing it, called the Fast
Fourier Transform (FFT), which was known to Gauss (1805) and was brought
to light in its current form by Cooley and Tukey <a class="reference internal" href="#rfb1dc64dd6a5-ct" id="id1">[CT]</a>.  Press et al. <a class="reference internal" href="#rfb1dc64dd6a5-nr" id="id2">[NR]</a>
provide an accessible introduction to Fourier analysis and its
applications.</p>
<p>Because the discrete Fourier transform separates its input into
components that contribute at discrete frequencies, it has a great number
of applications in digital signal processing, e.g., for filtering, and in
this context the discretized input to the transform is customarily
referred to as a <em>signal</em>, which exists in the <em>time domain</em>.  The output
is called a <em>spectrum</em> or <em>transform</em> and exists in the <em>frequency
domain</em>.</p>
</div>
<div class="section" id="implementation-details">
<h2>Implementation details<a class="headerlink" href="#implementation-details" title="Permalink to this headline">¶</a></h2>
<p>There are many ways to define the DFT, varying in the sign of the
exponent, normalization, etc.  In this implementation, the DFT is defined
as</p>
<div class="math">
<p><img src="../_images/math/192fb50712b8530c868a329a12a27077c012daa9.svg" alt="A_k =  \sum_{m=0}^{n-1} a_m \exp\left\{-2\pi i{mk \over n}\right\}
\qquad k = 0,\ldots,n-1."/></p>
</div><p>The DFT is in general defined for complex inputs and outputs, and a
single-frequency component at linear frequency <img class="math" src="../_images/math/5b7752c757e0b691a80ab8227eadb8a8389dc58a.svg" alt="f"/> is
represented by a complex exponential
<img class="math" src="../_images/math/db128dc25c145961513f66c7eecb26c4c4135dcc.svg" alt="a_m = \exp\{2\pi i\,f m\Delta t\}"/>, where <img class="math" src="../_images/math/b4ed9c2e208e08edeca8b1550ec0840acd090276.svg" alt="\Delta t"/>
is the sampling interval.</p>
<p>The values in the result follow so-called “standard” order: If <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">=</span>
<span class="pre">fft(a,</span> <span class="pre">n)</span></code>, then <code class="docutils literal notranslate"><span class="pre">A[0]</span></code> contains the zero-frequency term (the sum of
the signal), which is always purely real for real inputs. Then <code class="docutils literal notranslate"><span class="pre">A[1:n/2]</span></code>
contains the positive-frequency terms, and <code class="docutils literal notranslate"><span class="pre">A[n/2+1:]</span></code> contains the
negative-frequency terms, in order of decreasingly negative frequency.
For an even number of input points, <code class="docutils literal notranslate"><span class="pre">A[n/2]</span></code> represents both positive and
negative Nyquist frequency, and is also purely real for real input.  For
an odd number of input points, <code class="docutils literal notranslate"><span class="pre">A[(n-1)/2]</span></code> contains the largest positive
frequency, while <code class="docutils literal notranslate"><span class="pre">A[(n+1)/2]</span></code> contains the largest negative frequency.
The routine <code class="docutils literal notranslate"><span class="pre">np.fft.fftfreq(n)</span></code> returns an array giving the frequencies
of corresponding elements in the output.  The routine
<code class="docutils literal notranslate"><span class="pre">np.fft.fftshift(A)</span></code> shifts transforms and their frequencies to put the
zero-frequency components in the middle, and <code class="docutils literal notranslate"><span class="pre">np.fft.ifftshift(A)</span></code> undoes
that shift.</p>
<p>When the input <em class="xref py py-obj">a</em> is a time-domain signal and <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">=</span> <span class="pre">fft(a)</span></code>, <code class="docutils literal notranslate"><span class="pre">np.abs(A)</span></code>
is its amplitude spectrum and <code class="docutils literal notranslate"><span class="pre">np.abs(A)**2</span></code> is its power spectrum.
The phase spectrum is obtained by <code class="docutils literal notranslate"><span class="pre">np.angle(A)</span></code>.</p>
<p>The inverse DFT is defined as</p>
<div class="math">
<p><img src="../_images/math/7de8229bc434f3dfae9b186316c1721563f46ef7.svg" alt="a_m = \frac{1}{n}\sum_{k=0}^{n-1}A_k\exp\left\{2\pi i{mk\over n}\right\}
\qquad m = 0,\ldots,n-1."/></p>
</div><p>It differs from the forward transform by the sign of the exponential
argument and the default normalization by <img class="math" src="../_images/math/cb0504c49ba210e1759afd1e6491992917aa174a.svg" alt="1/n"/>.</p>
</div>
<div class="section" id="type-promotion">
<h2>Type Promotion<a class="headerlink" href="#type-promotion" title="Permalink to this headline">¶</a></h2>
<p><a class="reference internal" href="#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a> promotes <code class="docutils literal notranslate"><span class="pre">float32</span></code> and <code class="docutils literal notranslate"><span class="pre">complex64</span></code> arrays to <code class="docutils literal notranslate"><span class="pre">float64</span></code> and
<code class="docutils literal notranslate"><span class="pre">complex128</span></code> arrays respectively. For an FFT implementation that does not
promote input arrays, see <a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/fftpack.html#module-scipy.fftpack" title="(in SciPy v1.4.1)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.fftpack</span></code></a>.</p>
</div>
<div class="section" id="normalization">
<h2>Normalization<a class="headerlink" href="#normalization" title="Permalink to this headline">¶</a></h2>
<p>The default normalization has the direct transforms unscaled and the inverse
transforms are scaled by <img class="math" src="../_images/math/cb0504c49ba210e1759afd1e6491992917aa174a.svg" alt="1/n"/>. It is possible to obtain unitary
transforms by setting the keyword argument <code class="docutils literal notranslate"><span class="pre">norm</span></code> to <code class="docutils literal notranslate"><span class="pre">&quot;ortho&quot;</span></code> (default is
<em class="xref py py-obj">None</em>) so that both direct and inverse transforms will be scaled by
<img class="math" src="../_images/math/5c1e385692393dc6b3247fca6807fdd874ab311e.svg" alt="1/\sqrt{n}"/>.</p>
</div>
<div class="section" id="real-and-hermitian-transforms">
<h2>Real and Hermitian transforms<a class="headerlink" href="#real-and-hermitian-transforms" title="Permalink to this headline">¶</a></h2>
<p>When the input is purely real, its transform is Hermitian, i.e., the
component at frequency <img class="math" src="../_images/math/0edac3a6b0973196c5005c560d87e755c0ea5c86.svg" alt="f_k"/> is the complex conjugate of the
component at frequency <img class="math" src="../_images/math/a9527fff1f9301ecdeff1120a4a4ddde090caac0.svg" alt="-f_k"/>, which means that for real
inputs there is no information in the negative frequency components that
is not already available from the positive frequency components.
The family of <a class="reference internal" href="generated/numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a> functions is
designed to operate on real inputs, and exploits this symmetry by
computing only the positive frequency components, up to and including the
Nyquist frequency.  Thus, <code class="docutils literal notranslate"><span class="pre">n</span></code> input points produce <code class="docutils literal notranslate"><span class="pre">n/2+1</span></code> complex
output points.  The inverses of this family assumes the same symmetry of
its input, and for an output of <code class="docutils literal notranslate"><span class="pre">n</span></code> points uses <code class="docutils literal notranslate"><span class="pre">n/2+1</span></code> input points.</p>
<p>Correspondingly, when the spectrum is purely real, the signal is
Hermitian.  The <a class="reference internal" href="generated/numpy.fft.hfft.html#numpy.fft.hfft" title="numpy.fft.hfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">hfft</span></code></a> family of functions exploits this symmetry by
using <code class="docutils literal notranslate"><span class="pre">n/2+1</span></code> complex points in the input (time) domain for <code class="docutils literal notranslate"><span class="pre">n</span></code> real
points in the frequency domain.</p>
<p>In higher dimensions, FFTs are used, e.g., for image analysis and
filtering.  The computational efficiency of the FFT means that it can
also be a faster way to compute large convolutions, using the property
that a convolution in the time domain is equivalent to a point-by-point
multiplication in the frequency domain.</p>
</div>
<div class="section" id="higher-dimensions">
<h2>Higher dimensions<a class="headerlink" href="#higher-dimensions" title="Permalink to this headline">¶</a></h2>
<p>In two dimensions, the DFT is defined as</p>
<div class="math">
<p><img src="../_images/math/ed091ae5f8395421f58d2a9cbb2b872d4747a7ea.svg" alt="A_{kl} =  \sum_{m=0}^{M-1} \sum_{n=0}^{N-1}
a_{mn}\exp\left\{-2\pi i \left({mk\over M}+{nl\over N}\right)\right\}
\qquad k = 0, \ldots, M-1;\quad l = 0, \ldots, N-1,"/></p>
</div><p>which extends in the obvious way to higher dimensions, and the inverses
in higher dimensions also extend in the same way.</p>
</div>
<div class="section" id="references">
<h2>References<a class="headerlink" href="#references" title="Permalink to this headline">¶</a></h2>
<dl class="citation">
<dt class="label" id="rfb1dc64dd6a5-ct"><span class="brackets"><a class="fn-backref" href="#id1">CT</a></span></dt>
<dd><p>Cooley, James W., and John W. Tukey, 1965, “An algorithm for the
machine calculation of complex Fourier series,” <em>Math. Comput.</em>
19: 297-301.</p>
</dd>
<dt class="label" id="rfb1dc64dd6a5-nr"><span class="brackets"><a class="fn-backref" href="#id2">NR</a></span></dt>
<dd><p>Press, W., Teukolsky, S., Vetterline, W.T., and Flannery, B.P.,
2007, <em>Numerical Recipes: The Art of Scientific Computing</em>, ch.
12-13.  Cambridge Univ. Press, Cambridge, UK.</p>
</dd>
</dl>
</div>
<div class="section" id="examples">
<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
<p>For examples, see the various functions.</p>
</div>
</div>


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